Lexis cubes 1: From maps of space to maps of time

Jon Minton16 - 138 Jon Minton

A Lexis surface is a Cartesian mapping of three attributes to three dimensions:

  • year (or another measure of absolute time) to the x axis,
  • age (or another measure of relative time) to the y axis,
  • a third variable, which co-varies with year and age, to the z axis.

Put another way: a Lexis surface is a way of visualising temporal change as if it were spatial change, of thinking about time as if it were space: of absolute time as if it were latitude, relative time as if it were longitude, and a third variable as if it were a height above sea level.

Lexis Surfaces and map-making
A Lexis surface, like a spatial surface, is a surface, and for centuries the challenge of communicating the topography of spatial surfaces has been the challenge of producing meaningful maps of the world. The historic, strategic, political and military value of effectively conveying territorial topographies – features natural and man-made – is fossilised in the name of the UK’s national mapping agency, the Ordnance Survey, which owes its existence to the Jacobite rebellion of 1745, and the long history of conflict and compromise shared between England and Scotland, when ‘Bonnie Prince Charlie’ gained the support of clansmen in the Scottish highlands in his bid to regain the British throne.  As the name implies, the highlands are anything but flat, and so for armies laden with rations, muskets and ammunition, maps accurately showing which paths followed valleys, and which led up mountains or into lochs, could make the difference between victory and defeat. Over many centuries, maps of space, meeting the challenge of how to represent a three dimensional surface on a two dimensional page or parchment, have not just represented the world, but also shaped it.

Figures 1A&B

Figure 1A (left): Topographical map of central Scotland. (Image credit: https://simple.wikipedia.org/wiki/Highland_Boundary_Fault#/media/File:Scotland.central.topo.jpg -CC BY SA.)
Figure 1B (right): Demographic map of Scotland. The legend on the right indicates the base 10 log mortality risk (i.e. the ‘number of zeroes’ in the risk of dying in the next 12 months: 1 means one-in-10, 2 means one-in-100, 3 means one-in-1000, and so on). Source: Human Mortality Database

Demographic map-making
The processes and techniques of map-making can also be applied to Lexis surfaces. In 2013 I, along with Laura Vanderbloemen and Danny Dorling, published an article in the International Journal of Epidemiology showing a series of maps of all-cause mortality in England and Wales from 1841 onwards, as well as some other European countries, in which the ‘height’ of the Lexis surface was determined by death rates for each single year and age in single years, meaning tens of thousands of values could be represented on a single image. Like on an orienteering map, contour lines were used to indicate paths along the surface in which the ‘height’ was constant. During the first half of the twentieth century, and despite the deaths of the two world wars, which appeared as shards on the surface, many of these contour lines, representing a given risk of death, moved upwards a risk delayed, to be faced a few years later in life. From the 1950s onwards the hurdles have still continued to rise steadily and inexorably, suggesting longevity is still increasing: a cause for quiet celebration for all but pension fund managers and economists who only see the world through the lens of the working age dependency ratio. Some of the many potential applications for Lexis surface visualisations have been illustrated elsewhere, and since applied to understanding phenomena such as sex ratios in mortality in the USA, and to alcohol-based mortality in Scotland.

From mapping to seeing Lexis surfaces
Rather than simply mapping the Lexis surfaces, however, modern computer technology – and historically a great imagination and a steady hand has allowed the surfaces to be seen and explored directly. After some experimentation with computer generated images, rendering the surfaces as interactive virtual structures, with the support of the University of Glasgow’s Chancellor’s Fund, I have since used 3D printing to turn many of these surfaces into real, physical structures. In total more than 40 separate data cubes, each 8cm by 8cm by 8cm in size, showing log mortality rates, population sizes, and fertility rates from dozens of countries who submitted data to the Human Mortality Database and Human Fertility Database have been produced. Even a single Lexis cube, however, contains a huge amount of data, and can take a while to understand and interpret effectively. Because of this, in the next blog post I will discuss just one cube in detail.


Read more:

Minton J. Lexis Cubes 2 – Case-study: Log mortality for males in Finland, 1878 to 2012. International Journal of Epidemiology Blog 2016: URL: https://ije-blog.com/2016/06/27/lexis-cubes-2-case-study-log-mortality-for-males-in-finland-1878-to-2012/.

Minton J, Vanderbloemen L, Dorling D. Visualizing Europe’s demographic scars with coplots and contour plots. Int J Epidemiol 2013;42:1164–76. doi:10.1093/ije/dyt115.

Vanderbloemen L, Minton J, Dorling D. Visualizing sex differences in mortality, USA, 1933-2010. J Epidemiol Community Health 2016; doi:10.1136/jech-2014-205226.

McCartney G, Bouttell J, Craig N, et al. Explaining trends in alcohol-related harms in Scotland, 1991–2011 (I): the role of incomes, effects of socio-economic and political adversity and demographic change. Public Health 2016;132:13–23. doi:10.1016/j.puhe.2015.12.013.

Minton J. Logs, lifelines, and lie factors. Environ Plan A 2013;45:2539–43. doi:10.1068/a130208g.

Barbieri M, Wilmoth JR, Shkolnikov VM et al. Data Resource Profile: The Human Mortality Database (HMD). Int J Epidemiol 2015;44:1549-1556. doi:10.1093/ije/dyv105.

Human Mortality Database. Univ. California, Berkeley (USA), Max Plank Inst. Demogr. Res. 2011.

Goldstein JR, Shkolnikov VM, Sobotka T. Human Fertility Database. 2013.


Dr Jon Minton is an AQMEN Research Fellow based in the College of Social Sciences at the University of Glasgow. His background and current work straddles the social and health sciences, with a common methodological focus on complex data visualisation and data science, including promoting open and reproducible research. To this end, a large series of files for 3D printing data surfaces based on data from the Human Mortality Database and Human Fertility Database, and the R code used to produce them, are made available from the following location: https://github.com/JonMinton/Statistical_Sculpture/tree/master/stl/individual

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